Abstract
Keywords
1. Introduction
Terahertz (THz) waves have high penetration through non-metallic and non-polar materials, and they have low photon energy, so are safer for biological targets and human operators. Because of these special properties, both the two-dimensional (2D) and three-dimensional (3D) THz imaging techniques have gained widespread attention in biomedical fields[
A number of methods for 3D imaging with THz radiation have also been proposed and demonstrated. One of main modalities is the THz computed tomography (CT)[
Diffraction tomography (DT) is one of the common tomography methods that accounts for the diffraction effect[
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In this Letter, we proposed a CW-THz DT system by using a coherent single-frequency THz laser and the array detector to directly record the off-axis digital hologram. The setup is simple and easy to operate. By rotating the object, the 2D scattered fields of the object at each rotation angle can be obtained by the digital holographic reconstruction method. With regard to the 3D reconstruction algorithm of the THz DT, to the best of our knowledge, the filtered backpropagation (FBPP) method is first introduced to produce the volume data in the space domain and to achieve high-quality 3D RI measurement. Compared with the regular FDI algorithm, there is no need to do complex interpolation in the Fourier domain, and the experimental results demonstrate the validity of the method with high fidelity.
2. Principle of the Filtered Backpropagation Method
On the assumption of the scalar wave diffraction theorem, consider a monochromatic plane wave incident upon a scattering object, whose physical quantity is termed the scattering potential . The resultant total field [incident field plus scattered field ] satisfies the inhomogeneous Helmholtz equation as[
In order to solve Eq. (1), we can make use of Green’s function and apply the Rytov approximation, so the scattered field can be obtained as[
Then, a direct relation between the Fourier transform (FT) of the scattered field and the FT of the scattering potential of the object can be deduced, which is commonly called the FDT[
To reconstruct exact directly in the space domain[
In Eq. (5), there are two exponential terms inside the integral formula. The first one is a transfer function, which is depth-dependent due to the parameter . The second one is a factor to make the whole integral form a 2D inverse FT over and . The whole integral formula can be interpreted as propagating the scattered field along the direction, which is similar to the diffraction propagation. In order to implement the numerical calculation of Eq. (5), the scattered field can be propagated into sections in the space containing the object, and the interval between adjacent sections is denoted as , which can be flexibly set. Thus, volume data in the space domain is produced, and the scale of the depth is . Finally, the transformation of coordinates, illustrated as Eq. (6), is performed for 3D volumes, which are then added to the reconstruction volume in the sum over all rotation angles . Correspondingly, the lateral and axial resolutions of the DT system are related to the resolution of the complex amplitude distribution of a single projection and the number of projections[
It is noted that the FBPP algorithm generally requires scattered data measured from view angles in [] for exact reconstruction of a complex-valued object function. Compared to the direct linear inversion (the FDI algorithm) based on Eq. (4), there is no need to do the complex interpolation processing in the frequency domain, which will cause large computational errors and produce artifacts. The error-prone frequency interpolation can be avoided by applying the FBPP algorithm, which is theoretically equivalent to the FDI.
Practically, in order to achieve the complex diffracted fields (including the amplitude and phase distributions) of the object for one illumination angle , the off-axis digital holographic or the phase-shifting interferometry method can be employed. The off-axis digital holographic method is used in this Letter. The filtering frequency spectrum is applied to the digital holograms to obtain and , which are with and without the object. The scattered field can be calculated by Eq. (3).
3. Experimental Setup and Results
An experimental CW THz DT system based on off-axis Fresnel digital holography was built as depicted in Fig. 1 to collect the scattered field of the sample with different rotation angles. The THz source was an optically pumped far-IR gas laser (295-FIRL; Edinburgh Instruments Ltd., UK) with a central wavelength of 118.83 µm (2.52 THz) and a maximum power of 500 mW. The emitted THz beam was expanded and collimated by two off-axis parabolic mirrors (PM1, ; PM2, ) and then divided into the object beam and the reference beam by a beam splitter (BS). The object was attached to a metal holder for fixing on a rotational stage (PRMTZ8/M; Thorlabs Inc., USA). The interference fringes generated by the two beams formed the hologram, which was recorded by a pyroelectric detector (Pyrocam IV, pixel size , pixel pitch ). The distance from the object to the detector was . To enhance the contrast of the interference fringes, 500 frames were recorded at a chopping frequency of 50 Hz and accumulated via Gaussian fitting. In this experimental setup, the theoretical resolutions are about 0.3 mm.
Figure 1.Schematic of the experimental configuration of continuous-wave terahertz diffraction tomography (CW THz DT). PM1 and PM2, off-axis parabolic mirrors; BS, THz beam splitter; M, gold-coated mirror; RS, rotational stage.
To validate the proposed method, we first used a single polystyrene (PS) foam sphere with a diameter of 7.44 mm. The RI of the PS foam was at 2.52 THz, as measured by the THz-TDS system (TAS7400SU; Advantest, Japan). In the experiments, the sample was rotated gradually by 360° at angular intervals of 3°, and thus 120 holograms with the object were recorded successively, followed by one background hologram without the object. Figures 2(a1) and 2(b1) show the holograms with and without the object, respectively, at the rotation angle of 0°, and Figs. 2(c1) and 2(d1) show the reconstructed amplitude and wrapped phase images of Fig. 2(a1). The least-squares method is used to obtain the unwrapped phase image[
Figure 2.Reconstructed results of digital holography for single polystyrene (PS) foam sphere: (a1), (b1) holograms with and without the object at 0°; (c1), (d1) reconstructed amplitude and wrapped phase images of (a1); (a2)–(d2) reconstructed phase images at 0°, 45°, 90°, and 180°; (e1) phase profiles of the black dotted line in (a2)–(d2); (e2) result of (e1) after alignment.
The reconstructed complex amplitudes by digital holography with various angles were then processed to obtain the scattered field. The scattering potential distributions of the sample were reconstructed by using Eq. (5), where is set to be 80 µm, and N is equal to 320. The RI distributions were finally achieved by using Eq. (2). In our experiment, the sample was supported by a metal rod whose large RI led to incorrect reconstructed values in these areas, so the support-rod part has been removed from the reconstructed 3D RI results below.
Figures 3(a1)–3(c1) are the reconstructed results by the FDI method, showing the obtained RI distributions at cross sections , , and , respectively. It is seen that there is some artifacts error inside, especially in the background region, and the fluctuation of the value is relatively serious. The reconstructed RI distributions by the FBPP method are shown in Figs. 3(a2)–3(c2), which show much-improved fidelity of the reconstructed RI tomograms. Figure 3(d) shows the differences of the RI profiles of the PS foam sphere between the results obtained by the FBPP method and the FDI method. The red line represents the ideal RI value of the PS foam material, which is measured by the THz-TDS system. As can be seen, the RI value obtained by the FBPP method is more accurate, where the average value is , and the error is only 0.16%, thereby verifying the effectiveness of the proposed method. Furthermore, the average diameter of the reconstructed foam sphere is , which corresponds to a reconstruction error of 0.6% compared with the true value. We can also analyze the reconstruction quality of the FBPP and FDI methods from the view of 3D Fourier space, as shown in Fig. 4. Figures 4(a) and 4(b) are the amplitude of the spectrum distribution of FDI and FBPP on the logarithmic scale along the , and cross sections, respectively. For the FDI method, the number of missing spectrum points produced by the reconstruction algorithm is more than that of FBPP, as indicated by the green arrows. It leads to the reconstructed RI distribution being worse, as shown in Figs. 3(a1)–3(c1). Meanwhile, both FBPP and FDI have the same missing spectrum points caused by the “missing apple core” problem, as indicated by the red arrows.
Figure 3.Reconstructed refractive index (RI) profiles of DT for a single foam sphere: (a1)–(c1), (a2)–(c2) 3D RI profiles at cross sections x–y, y–z, and x–z by FDI and FBPP algorithm, respectively; (d) RI profiles of the red dotted line in (c1) and (c2), and ideal values obtained by the THz-TDS system.
Figure 4.Amplitude of the spectrum distribution of (a) FDI and (b) FBPP on the logarithmic scale along the fx–fy, fx–fz, and fy–fz cross sections, respectively.
To demonstrate further the applicability of THz DT on non-axisymmetric samples, two glued PS foam spheres were placed horizontally on the rotary stage, and then the sample was rotated around the gravity center of one sphere to obtain the scattered field with various rotation angles. The reconstructed RI distributions of the foam spheres at cross sections , and and the 3D distributions are shown in Figs. 5(a)–5(d), respectively. From Visualization 1, the 3D RI distributions are presented visually. The average RI value of the two reconstructed foam spheres is , and the difference is 0.42% compared with the value obtained by the THz-TDS system. Note that the RI distributions are not as uniform compared with the previous single foam sphere. There are some fringe patterns. Because multiple scattering exists when the THz beam propagates through the two foam spheres, the Rytov approximation is not well satisfied, and the reconstruction error is increased. To solve this, one possible way is to build the forward propagation model including the multiple scattering effects and apply an iterative DT reconstruction algorithm to improve the quality of 3D reconstructed RI distributions[
Figure 5.RI distributions of DT for two foam spheres: (a)–(c) 3D RI profiles at cross sections x–y, y–z, and x–z; (d) volume rendering of 3D RI profiles (see Visualization 1).
4. Summary and Discussion
In summary, we have realized CW THz DT combined with digital holography, and the digital holograms are recorded directly by the array detector at various angles through rotating the samples. The configuration is simple and easy to operate. With regard to the 3D reconstruction algorithm, the FBPP algorithm is adopted to achieve 3D RI distributions of the PS foam spheres. The average RI value has only 0.16% difference from the RI value measured by the THz-TDS system. The reconstructed results have high fidelity compared with results obtained by the FDI algorithm. It verifies the feasibility of the proposed method.
Note that in the reconstructed RI tomograms in Figs. 3 and 5, some values are quite different from their surroundings along the Y axis, which is the result of the “missing apple core” problem. It can be improved by the iterative algorithm with non-negative constraints. Furthermore, to promote THz DT further, RI matching or the multiple scattering non-linear model must be applied to reconstruct high-RI samples. Our view is that THz DT can be an effective method for non-destructive testing and quantitative measurement of the complex RI of complex samples in the future, which can be combined with other THz wide-field phase-contrast imaging methods.
References
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